Calculate the volumetric flow of a fluid

[CFXMSC]

How could I calculate the volumetric flow of a fluid due to the weight of a piston in a cylinder?

At first moment i I thought the Hagen-Poiseuille equation, but I'm not sure. Any suggestions?

 

[CFXMSC]

For Duct

Continuity:

$\frac{\partial v}{\partial y}=0$

Momentum:

$\rho g_y-\frac{\partial p}{\partial y}+\mu\left(\frac{\partial^2v}{\partial x^2}+\frac{\partial^2v}{\partial y^2}\right)=0$

For Circular Pipe:

Continuity:

$\frac{\partial}{\partial r}(v_z)=0$

Momentum:

$g-\frac{1}{\rho}\frac{\partial p}{\partial z}+\nu\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial v_z}{\partial r}\right) \right]$=0